1. Field of the Invention
The invention relates generally to image reconstruction in a cone beam imaging system having a radiation source scan path that encircles an object, and more specifically to the use of a simplified 3D backprojection image reconstruction technique in a cone beam imaging system.
2. Description of the Prior Art
A filtered backprojection (FBP) cone beam image reconstruction technique is described by Kudo, H. and Saito, T., in their article entitled "Derivation and Implementation of a Cone-Beam Reconstruction Algorithm for Nonplanar Orbits", IEEE Trans. Med. Imag., MI-13 (1994) 196-211, incorporated herein by reference.
Briefly, the FBP technique consists of the following steps at each cone beam view (i.e., at each position of the radiation source as it scans about the object, and at which an imaging detector acquires a corresponding set of projection data):
1. Compute a 1D projection (i.e., line integral) of the measured cone beam image acquired on a detector plane 1 at each of a plurality of angles .theta.. This step is illustrated by FIG. 1A for a given angle .theta..sub.1 of a plurality of angles .theta., where the projection 2 at coordinates (r,.theta.) comprises the integrated values of the cone beam image 4 on detector plane 1 along plurality of parallel lines L(r,.theta.) that are normal to angle .theta., each line L being at an incremental distance r from an origin O. Generally, if the detector plane 1 comprises an N by N array of pixels, then the number of angles .theta. is typically given by .pi.N/2. PA1 2. Filter each 1D projection in accordance with a d/dr filter, resulting in a new set of values at each of the r,.theta. coordinates, such as shown by filtered projection 6 for the angle .theta..sub.1 in FIG. 1A. PA1 3. Normalize the filtered projections with a normalization function M(r,.theta.). Normalization is needed to take into account the number of times the plane of integration Q(r,.theta.) which intersects the source position and the line L(r,.theta.), intersects the scan path, since the data developed at each scan path intersection creates a contribution to the image reconstruction on the plane Q(r,.theta.). PA1 4. Backproject the filtered projection 6 from each angle .theta. into a 2D object space 7 which coincides with the detector plane 1. This step is illustrated by FIG. 1B, wherein lines 8 spread the value from each r,.theta. coordinate into 2D space 7 in a direction normal to each .theta.. PA1 5. Perform a 1D d/dt filtering of the backprojection image formed in 2D space 7 by step 4. The 1D filtering is performed in the direction of the scan path, i.e., along lines 10, where t points in the direction of the scan path. PA1 6. Perform a weighted 3D backprojection of the resulting data in 2D space 7 (i.e., from each pixel in the detector) onto a plurality of sample points P in a 3D object volume 12. The density assigned to each point P is weighted by the inverse of the square of the distance between the point and the apparent x-ray source (see Equation (59) of the forenoted Kudo et al article).
The above prior art procedure will be referred to hereinafter as the 6-step process. It is assumed in this process that the entire cone beam image of the object is captured on the detector of the imaging system. Consider a plane Q(r,.theta.), which intersects the object, formed by the source and the line L(r,.theta.) on the detector at angle .theta. and at a distance r from the origin. Ignoring the function M(r, .theta.), the operations 1 through 6 compute the contribution to the reconstructed object density on the plane Q(r,.theta.) from the x-ray data illuminating the plane and its immediate vicinity. Since the 6-step process is detector driven, a contribution from the data illuminating the plane is computed every time the plane intersects the scan path and thus is illuminated by the x-ray beam. Consequently, the function M(r,.theta.) is used after the filter function in step 2 to normalize the results. Normalization is particularly undesirable since it requires pre-computing and storing a 2D array M(r,.theta.) for each source position along an imaging scan path. Since there are usually hundreds, if not thousands of source positions, this type of normalization is both computationally intensive and resource (computer memory) expensive. In the Kudo et al article, however, at page 203 it is noted that in the special case where the scan path is a circle, steps 1-5 can be simplified into a single convolution step, which essentially comprises ramp filtering the cone beam image in the direction of the scan path. This ramp filtering is equivalent to the well-known Feldkamp algorithm for a single circular orbit, such algorithm being described in particular detail in the article by L. A. Feldkamp, L. C. Davis, and J. W. Kress, entitled "Practical cone-beam algorithm" published in the J. Opt. Soc. Am. A. Vol. 1, 1984, pages 612-619 (see in particular the convolution function equations 15 and 16 at page 614, which describe the convolution function as: ##EQU1## incorporated by reference herein. The key to this simplification is that in the case of a circular scan path, the normalization function M(r,.theta.) is a constant, equal to 2. Consequently, the filtered projection at each r,.theta. that results after step 2, can merely be divided by 2 to compensate for the data redundancy.
It would be desirable to use this simplification for source scan paths other than a single circle, but the Kudo et al article does not provide any indication how this can be done, or that it is even possible.
Furthermore, since the above procedure assumes that the detector captures the entire cone beam image of the object at each view, it can not be applied to a cone beam imager having a short detector, i.e., one that only captures a portion of the cone beam image at each cone beam view. Thus, in its current form, the Kudo et al. FBP technique cannot be applied to a cone beam imager having a spiral scan path and employing a short detector.
It would be desirable to apply the Kudo et. al. image reconstruction processing simplification for a single circle scan imager, to a cone beam imaging system that not only has a spiral scan path, but also uses a short detector.